Probabilistic Optical and Radar Initial Orbit

Determination

Roberto Armellin1

University of Surrey, Guildford, GU2 7XH, United Kingdom

Pierluigi Di Lizia2

Politecnico di Milano, Milano, 20156, Italy

Future space surveillance requires dealing with uncertainties directly in the ini-

tial orbit determination phase. We propose an approach based on Taylor differential

algebra to both solve the initial orbit determination (IOD) problem and to map uncer-

tainties from the observables space into the orbital elements space. This is achieved

by approximating in Taylor series the general formula for probability density function

(pdf) mapping through nonlinear transformations. In this way the mapping is ob-

tained in an elegant and general fashion. The proposed approach is applied to both

angles-only and two position vectors IOD for objects in LEO and GEO.

I. Introduction

The total number of active satellites, rocket bodies, and debris larger than 10 cm is currently

about 20,000. Considering all resident space objects larger than 1 cm this rises to an estimated

minimum of 500,000 objects. Latest generation sensor networks will produce millions of observations

per day [1, 2]. Identifying observations belonging to the same object will be one of the main

challenges of orbit determination. One requirement to perform reliable data association is to have

realistic uncertainties for initial orbit solutions, which could also be used to initialize Bayesian

estimators for orbit refinement [3]. Meeting this requirement demands two main steps: solving the

1 Senior Lecturer, Surrey Space Centre, BA Building; r.armellin@surrey.ac.uk.2 Assistant Professor, Department of Aerospace Science and Technology, via La Masa 34; pierluigi.dilizia@polimi.it.

1

initial orbit determination (IOD) problem, and mapping the error distributions from observations

to the initial orbit solutions. The second step is the main focus of this work.

In IOD, the number of observations is equal to the number of unknowns, and thus a nonlinear

system of equations is to be solved. Moreover, simplified dynamical models are often used (e.g.

Keplerian motion) and measurement errors are not taken into account (the problem is deterministic).

Based on the set of available measurements, different IOD problems can be framed. In this work

we consider two cases: angles-only and two position vectors IOD. In angles-only IOD a resident

space object (RSO) is observed three times with optical sensors on a single passage. Angles-only

IOD is an old problem. Gauss’ [4] and Laplace’s [5] methods are commonly used to determine a

Keplerian orbit that fits with three astrometric observations. These methods have been revisited and

analyzed by a large number of authors [6–8], and new ones introduced more recently. The Double

r-iteration technique of Escobal [9] and the approach of Gooding [10] are two examples of angles-

only IOD methods for RSO, whereas a procedure to correct the deterministic solution provided by

the classical IOD algorithms has been introduced by Hussein et al. [11] to improve the performance

of observation-to-observation association schemes. The admissible region methods, which were

originally developed by Milani et al. [12], provide an alternative approach to the representation of the

solution of the IOD problem [13, 14]. In the two position vectors problem, two range measurements

are obtained with a radar together with the information on instrument pointing angles. In this

case, the classical Lambert problem can be formulated [15]. Although this problem was solved more

than 200 years ago, many researchers are still working on devising robust and efficient resolution

procedures [16, 17].

In general, the solution of both angles-only and two position vectors IOD is numerical and

thus an explicit function that relates the observations with the object orbital parameters is not

available. The classical methods proposed in the literature to map observation statistics into initial

orbit statistics rely on linearizations [18]. However, the validity of the linearity assumption is often

called into question due to the inherent nonlinearity of the mapping from the measurements space

to the orbital space. Consequently, alternative methods have been recently devised to deal with

nonlinearities, such as the use of particle based approaches [19–22] or the recent technique of the

2

transformation of variables to map probability density functions from a specified domain into a

different domain [23, 24]. The method proposed in this work shares the same objective of the latter,

with the difference that differential algebra (DA) is used to compute a semi-analytical representation

of the mapping between the initial and the desired domains.

In this work, the relation between the observations and the object orbital parameters is ap-

proximated in terms of its high order Taylor expansion by using DA [25]. DA is used to solve

the nonlinear equations associated to the IOD problem as well as to expand their solution in Tay-

lor series with respect to the measurements. The resulting high-order map is referred to as the

observations-to-solution (O2S) map. The O2S map can be profitably used to map analytically sam-

ples from the observation space into the RSO orbital parameters space, thus enabling for example

efficient Monte Carlo (MC) simulations. In addition, for given statistical properties of the measure-

ments, the availability of a Taylor approximation of the O2S allows one to obtain the statistical

moments of the states by computing the expectation of its monomials, thus avoiding the need of

samples [26, 27]. More importantly, the O2S map (and the Taylor representation of its inverse,

the S2O map) can be manipulated to analytically represent the probability density function (pdf)

of the IOD solution for any arbitrary pdf of the measurements. This is achieved by expanding in

Taylor series the determinant of the Jacobian of the O2S map and using well known formula for pdf

transformation [28, 29]. This approach is independent on the representation of the dynamics and

thus the Keplerian approximation can be removed.

The paper is organised as follows. An introduction on DA is given first, together with the

algorithms for the expansion of the solution of implicit equations and the solution of a system of

ODE. This section ends with the description of the procedure to select the expansion order. Section

III describes the use of DA for mapping a pdf in a general fashion, using the Cartesian to polar

transformation as an illustrative case. The method is then detailed in Sec. IV for the specific

application of angles-only and two position vectors IOD. The optical observation of GEO spacecraft

and the radar observation of an RSO in LEO are used as test cases in Sec. V to investigate the

method’s features and performance. Some final remarks conclude the paper.

3

II. Differential Algebra tools

DA supplies the tools to compute the derivatives of functions within a computer environment

[25]. More specifically, by substituting the classical implementation of real algebra with the imple-

mentation of a new algebra of Taylor polynomials, any deterministic function f of v variables that is

Ck+1 in the domain of interest [−1, 1]v (these properties are assumed to hold for any function dealt

with in this work) is expanded into its Taylor polynomial up to an arbitrary order k with limited

computational effort. In addition to basic algebraic operations, operations for differentiation and

integration can be easily introduced in the algebra, thusly finalizing the definition of the differential

algebraic structure of DA [30, 31]. Similarly to algorithms for floating point arithmetic, various al-

gorithms were introduced in DA, including methods to perform composition of functions, to invert

them, to solve nonlinear systems explicitly, and to treat common elementary functions [32]. The

DA used for the computations in this work was implemented in the softwares COSY INFINITY [33]

and DACE [34]. The reader may refer to [35] for the DA notation adopted throughout the paper.

A. High-order expansion of the solution of ODE

An important application of DA is the automatic computation of the high order Taylor expansion

of the solution of an ODE with respect to either the initial conditions or any parameter of the

dynamics [32, 35]. This can be achieved by replacing the classical floating point operations of the

numerical integration scheme, including the evaluation of the right hand side, by the corresponding

DA-based operations. This way, starting from the DA representation of the initial condition x0, the

DA-based ODE integration supplies the Taylor expansion of the flow in x0 at all the integration

steps, up to any final time tf . Any explicit ODE integration scheme can be adapted to work in the

DA framework in a straightforward way. For the numerical integrations presented in this paper, a

DA version of a 7/8 Dormand-Prince (8-th order solution for propagation, 7-th order solution for

step size control) Runge-Kutta scheme is used. The main advantage of the DA-based approach is

that there is no need to write and integrate variational equations to obtain high order expansions of

the flow. It is therefore independent on the particular right hand side of the ODE and the method

is quite efficient in terms of computational cost.

4

B. Expansion of the solution of parametric implicit equations

Well-established numerical techniques (e.g., Newton’s method) exist to compute numerically

the solution of an implicit equation

h(y) = 0, (1)

with h : Rn → Rn. Suppose the vector function h depends explicitly on a vector of parameters x,

which yields the parametric implicit equation

h(y,x) = 0. (2)

The solution of (2) is the function y = f(x) that solves the equality for any value of x.

DA techniques can effectively handle the previous problem by representing f(x) in terms of its

Taylor expansion with respect to x. This result is achieved by applying partial inversion techniques

as detailed in Di Lizia et al. [35]. The final result is

y = T kf (x), (3)

which is the k-th order Taylor expansion of the solution of the implicit equation. For every value

of x, the approximate solution of h(y,x) = 0 can be easily computed by evaluating the Taylor

polynomial (3). Apparently, the solution obtained by means of the polynomial map (3) is a Taylor

approximation of the exact solution of Eq. (2). The accuracy of the approximation depends on both

the order of the Taylor expansion and the displacement of x from its reference value.

The capability of expanding the solution of parametric implicit equations (PIE) is of key im-

portance for this work, as these appear in both Lambert’s and Kepler’s problems, used here to solve

the IOD problems and to map the resulting solution manifold forward in time when the two-body

approximation is adopted. More importantly, any IOD problem can be framed as a PIE prob-

lem. The role of the vector of parameters x in Eq. (2) is played by the measurements, whereas

y is the spacecraft state and h(y,x) is the function of the residuals. Within this framework, the

resulting polynomial T kf in Eq. (3) maps analytically any variation of the measurements into the

corresponding variation of the orbital state. Apparently, T kf is the O2S map defined in Sec. I.

5

1. Kepler’s problem

In Kepler’s problem the initial position and velocity vectors, r1, and v1 of an object are known

at an epoch t1. The problem consists in finding the position and velocity vectors of the body, r2

and v2, at epoch t2 in the two-body approximation. This is done by computing the Lagrangian

coefficient f, f t and g, gt at time t2 and the new object state as

r2 = f2r1 + g2v1

v2 = f t2r1 + gt2v1

(4)

For ellipses, the Langrangian coefficients are function of the eccentric anomaly difference E2 − E1

and assume the form [15]

f2 = 1−a

r1[1− cos (E2 − E1)]

g2 =aσ1

õ

[1− cos (E2 − E1)] + r1

√√√√√a

µsin(E2 − E1)

f t2 = −µa

r2r1sin(E2 − E1)

gt2 = 1−a

r2[1− cos(E2 − E1)] ,

(5)

where a is the semi-major axis of the orbit, µ the gravitational parameter, σ1 = (r1 · v1)/√µ, and

r2 = ||f2r1 + g2v1||. To evaluate the Lagrangian coefficients, the eccentric anomaly difference is

computed by solving

M2 −M1 = (E2 − E1) +σ1

√a

[1− cos(E2 − E1)]−

1−r1

a

sin(E2 − E1), (6)

where M is the mean anomaly defined as M =√µ/a3 = n(t − τ), in which n is the mean motion

and τ the time of passage through the perigee.

Typically (6) is solved with few Newton’s iterations. Let us assume the dependence of the

solution with respect to a set of parameters is of interest (e.g., with respect to the measurement

errors as it appears later in Sect. IV, when the Kepler problem is solved within the IOD algorithm

to obtain the O2S map). This dependence can be represented by a Taylor polynomial by applying

the algorithm of Sec. II B. When the two-body problem approximation is adopted, the integration

6

of Kepler’s dynamics can be substituted by the solution of Kepler’s problem, thus reducing the

computational time.

2. Lambert’s problem

In Lambert’s problem, the initial position, final position, and the time of flight between the two

positions are given. Solving Lambert’s problem defines the Keplerian orbit that connects the two

position vectors in the given time, allowing the calculation of the velocities at the initial and final

positions.

Lambert’s theorem states that the time of flight depends only on the semi-major axis of the

connecting arc, the sum of the two radii r1 + r2, and the distance between the initial and final

positions, i.e., the chord length c = ||r2 − r1|| [36]. The two radii and the chord length are already

known from the problem definition. The semi-major axis is the only unknown parameter, and it

follows that it is possible to write the transfer time as a function of the semi-major axis only:

∆t = f(a), (7)

all other variables in the equation should be known. This can be done as in [37], or the time of flight

can be written as a function of some other parameters such as p or ∆E. In the approach adopted

in this work, based on Battin’s algorithm [15], the nonlinear equation to be solved is

log(A(x)

)− log(∆t) = 0, (8)

in which

A(x) = g(x)3/2(α(x)− sinα(x)− β(x) + sinβ(x)

). (9)

The functions α(x) and β(x) are related to x via the relations

sin2 1

2α(x) =

s

2g(x)sin2 1

2β(x) =

s− c2g(x)

, (10)

with

g(x) =s

2(1− x2), (11)

7

and the semi-perimeter

s = (r1 + r2 + c)/2. (12)

The logarithm in (8) is adopted to facilitate the solution for x by means of standard Newton’s

iteration. Note that the relation between a and x is simply given by

a =s

2(1− x2). (13)

Once the nominal solution is available, the dependence of the solution on an arbitrary set of

parameters can be represented by a Taylor polynomial using the algorithm introduced in Section

II B. The expansion of the solution of Lambert’s problem with respect to observation uncertainties

forms the basis of the IOD solver for both optical and radar observations, as described in Sec. IVA

and IVB.

C. Selection of the expansion order

The selection of the expansion order is a crucial issue when using high order Taylor represen-

tation. The order must be selected so that the truncation error of the Taylor representation of

the function of interest is lower than a threshold that is deemed appropriate for the problem at

hand. In this work the function f to be approximated is a generic Ck+1 function that maps the

input domain into the desired output space (e.g. the O2S map, which can include reference frame

changes and propagations as explained in Sec. IVC and Sec. IVD). The input domain is typically

scaled such that it is described by a v-dimensional hypercube [−1, 1]v. It is not possible to select

a priori the expansion order k to guarantee that the truncation error meets the accuracy require-

ments. However, once the Taylor approximation of order k is computed, its truncation error can be

estimated. Thus, one can either decide to work at an expansion order k and then verify that the

Taylor representation is accurate enough (the approach followed in this work) or can progressively

increase the expansion order until the accuracy requirement is met. However, there are two limiting

factors on the maximum order to be considered. The first one is that the computational time of a

Taylor expansion increases exponentially with the expansion order and the problem dimension v.

The second one is that the convergence radius of a Taylor series can be smaller than the uncertainty

8

domain of interest. In both cases, the only option one can pursue is to split the initial domain in

subdomains, which is the idea behind the technique of automatic domain splitting introduced in

[38].

Three main methods exist for estimating the truncation error: a sample-based method, a

coefficient-based method, and a rigorous estimation method. The sample-based method involves

sampling the uncertainty domain and evaluating both the original function f and its Taylor ap-

proximation Tf . The truncation error is readily estimated as the maximum difference between the

value of the function and its Taylor approximation. The main drawback of this approach is that it

can be time consuming and, by requiring the multiple evaluation of the f , it can partially impair

the advantage of working with Taylor approximations. However, as the truncation error increases

with the distance from the Taylor expansion point, only few samples belonging to the boundary of

the uncertainty domain (e.g. the corners of the [−1, 1]v or just a subset of them in the direction of

maximum error growth) can be used to estimated the truncation error.

The coefficient-based method has the advantage that it relies on the analysis of the coefficients of

the Taylor approximation. Thus, it does not requires significant computational effort. The method

estimates the size of the k + 1 order terms of the Taylor polynomial based on an exponential fit of

the size of all the known non-zero coefficients up to order k. The exponential fit is chosen because

when the Taylor approximation is converging, the coefficients of the resulting polynomial expansion

will in fact decay exponentially. This is a direct consequence of Taylor’s Theorem. Given the k-th

order Taylor representation of the function f

Tf (x) =∑α

aαxα

written using multi-index notation, the size Si of the terms of order i is computed as the sum of the

absolute values of all coefficients of exact order i:

Si =∑|α|=i

|aα| .

We denote by I the set of indices i for which Si is non-zero. A least squares fit of the exponential

function

t(i) = A · exp(B · i)

9

is used to determine the coefficients A,B such that t(i) = Si, i ∈ I, is approximated optimally in

the least squares sense. The value of t(k + 1) is used to estimate the size Sk+1 of the truncated

order k + 1 of Tf . This value is representative of the accuracy of the Taylor approximation of

order k in the domain of interest. Thus, for a given problem, the order k to be used for the DA

computations is selected so that Sk+1 is lower than a prescribed tolerance. It has to be remarked

that this approach gives an accurate estimate of the truncation error when a sufficient number of

coefficients is available for the fit, i.e. for a sufficiently high value of k (say above 3). For more

details the reader is referred to [38].

Lastly, the rigorous estimation method involves the use of interval algebra to rigorously bound

the remainder term of the Taylor approximation, as implemented in the Taylor-model approach [39].

By being fully rigorous, this approach allows one to carry out validated computations, which can

be used for example for computer assisted proofs. However, the use of this approach in complex

engineering problems can require significant coding effort and can greatly increase the computational

cost. For this reason the Taylor-model approach is not used in this work.

III. Uncertainty mapping

Consider two n-dimensional random vectors x and y, related by the continuous one-to-one

transformations y = f(x) and x = g(y), and the probability density function p(x). The goal of

uncertainty mapping is to represent the statistics of the transformed variable y. This section first

introduces a general method to map the pdf. The method relies on the high order Taylor expansion

of the transformation functions f and g, as well as of their Jacobians. Then, in Sec. III B an

approach that exploits the availability of the Taylor representation of f to compute the k-th order

approximation of the statistical moments of the mapped pdf is presented. Finally, a simpler method

is introduced in Sec. III C, which exploits the same Taylor expansions to reduce the computational

time of classical MC simulations.

A. DA-based pdf mapping

Using the axiom of probability conservation and the rules of change of variables in multiple

integrals, the probability density function of y can be written as [28]

10

p(y) = p(g(y))

∣∣∣∣det∂g∂y∣∣∣∣ , (14)

in which det indicates the determinant. By substitution, Eq. (14) can be rewritten in terms of the

original variables x as

p(f(x)) = p(x)

/∣∣∣∣det∂f∂x∣∣∣∣ . (15)

The goal of the method is mapping the pdf of the observations into the pdf of the orbital

parameters. Within this framework, the observations are represented by the vector x with known

p(x). As anticipated in Sec. I, there is no explicit relation between the observations and the orbital

parameters in IOD. This means that the relation y = f(x) is not known analytically and must be

computed numerically by solving an implicit parametric relation in the form h(y,x) = 0. This has

always been the classical impediment to obtaining an analytical expression of the mapped pdf.

As explained in Section II B, DA can solve the parametric implicit equation to provide a poly-

nomial representation of f , i.e. of the O2S map. The resulting polynomial can also be inverted to

deliver the Taylor approximation of the S2O map

x = Tg(y), (16)

in which the superscript k has been omitted for the sake of a lighter notation.

Furthermore, the Taylor expansion of the determinant of the Jacobian of both f and g can be

easily obtained by carrying out the pertaining algebraic operations in the DA framework. Conse-

quently, the computation of p(y) can be obtained from Eq. (15) or Eq. (14) by exploiting the Taylor

expansion of the determinants, i.e.

p(y) = p (Tg(y)) T|det ∂g∂y |(y) (17)

and

p (Tf (x)) = p (x) 1

/T|det ∂f

∂x |(x). (18)

11

(a) Computation of p(y) for a given y using Eq. (17)

(b) Computation of p(Tf (x)) for a given x using Eq. (18)

Fig. 1: Schematic representation of the partial DA-based pdf mapping method.

In Eq. (17), for any given y, p(y) is computed by evaluating Tg(y) to obtain the corresponding x,

computing p(x) from the known pdf of the observations, and multiplying the resulting pdf by the

value of the Taylor expansion of the determinant of g at y (see Fig. 1(a)). Alternatively, in Eq.

(18), for any given x, the pdf at y = Tf (x) is computed by multiplying the corresponding p(x) by

the inverse of the value of the Taylor expansion of the determinant of f at x (see Fig. 1(b)).

This approach is labeled as partial DA-based pdf mapping method as, in both Eq. (17) and

Eq. (18), the pdf of y is obtained by multiplying the value of the pdf of x by a Taylor polynomial.

However, if p(x) is known analytically, then both p (Tg(y)) and p (x) can be evaluated in the DA

framework. The result is the Taylor expansion of p(x) with respect to y and x, respectively.

Consequently, Eqs. (17) and (18) deliver a full polynomial representation of the mapped pdf, i.e.

p(y) ≈ Tp(y)(y) = Tp(Tg(y))(y) T|det ∂g∂y |(y) (19)

12

and

p (Tf (x)) ≈ Tp(Tf (x))(x) = Tp(x)(x)/T|det ∂f

∂x |(x). (20)

In Eq. (19), for any given y, p(y) is computed by evaluating directly Tp(y)(y) at y. Alternatively,

in Eq. (20), for any given x, the pdf at y = Tf (x) is computed by evaluating directly Tp(Tf (x))(x)

at x. This second approach is referred to as full DA-based pdf mapping. Note that, since p(x) is an

asymptotic function, the accuracy of the Taylor expansions Tp(y)(y) and Tp(Tf (x))(x) will be limited

in this case, even when the computations are carried out at high orders.

B. Computation of statistical moments

The statistical moments of the mapped pdf can be computed in a deterministic way by comput-

ing the expectation of the analytical approximation of the O2S map Tf (x), following the procedure

already introduced in [27].

For a generic scalar random variable x with pdf p(x), the first four moments can be written as

µ = Ex

P = E(x− µ)2

γ =E(x− µ)3

σ3

κ =E(x− µ)4

σ4− 3,

(21)

where µ is the mean value, P is the covariance, γ and κ are the skewness and the kurtosis, respec-

tively, and the expectation value of x is defined as

Ex =

∫ +∞

−∞xp(x)dx. (22)

So, the moments of the transformed pdf can be computed by applying the multivariate form

of Eq. (21) to the Taylor expansion of the O2S map, Tf (x). The result for the first two moments

becomesµi = ETf (x)i =

∑p1+···+pn≤k

ci,p1...pnEδxp1

1 · · · δxpnn

P ij = E(Tf (x)i − µi)(Tf (x)j − µj) =∑

p1+···+pn≤k,q1+···+qn≤k

ci,p1...pncj,q1...qnEδxp1+q11 · · · δxpn+qn

n ,

(23)

13

where ci,p1...pn are the Taylor coefficients of the Taylor polynomial describing the i-th component of

Tf (x). Note that in the covariance matrix formula the coefficients ci,p1...pn and cj,q1...qn already in-

clude the subtraction of the mean terms. The coefficients of the higher order moments are computed

by implementing the required operations on the Taylor polynomials in the DA framework. The ex-

pectation values on the right side of Eq. (23) are function of p(x) (e.g. (Tf (x)i −µi)(Tf (x)j −µj)

for the second order moment). It follows that, if the initial distribution is known, all of the moments

of the transformed pdf p(y) can be calculated. The number of monomials for which it is necessary

to compute the expectation increases with the order of the Taylor expansion and, of course, with

the order of the moment we want to compute. Note that, at this time, no hypothesis on the initial

pdf has been made. Thus, the method can be applied independently of the considered variable

distribution.

However, the Gaussian hypothesis is commonly used to represent the distribution of mea-

surement noise. So, we can restrict to the case where x is a Gaussian random variable (GRV),

x ∼ N (µ,P ), in which µ is the mean vector and P the covariance matrix. An important property

of Gaussian distributions is that the statistics of a GRV can be completely described by the first

two moments. In case of zero mean, the expression for computing higher-order moments in terms

of the covariance matrix is due to Isserlis [40]. In physics literature, Isserlis’s formula is known as

the Wick’s formula.

Let s1 to sn be nonnegative integers, and s = s1 + s2 + · · · + sn. Then the Wick’s formula

suggests that

Exs11 xs22 . . . xsnn =

0, if s is odd

Haf(P ), if s is even

(24)

where Haf(P ) is the hafnian of P = (σij), which is defined as

Haf(P ) =∑p∈∏

s

s2∏i=1

σp2i−1,p2i, (25)

and∏s is the set of all permutations p of 1, 2, . . . , s satisfying the property p1 < p3 < p5 < . . . <

ps−1 and p1 < p2, p3 < p4, . . . , ps−1 < ps [41].

We observe that the expectation value terms of Eq. (23) can be computed using Eq. (24),

14

and the resulting moments can be used to describe the transformed pdf. As a result, if one is only

interested in the statistical moments of the mapped pdf, MC simulations are not necessary, and

this result represents an additional advantage of having an analytical approximation of the O2S

function.

C. DA-based Monte Carlo

The DA-based MC method requires only the expansion of the f function, obtained as explained

in Sec. II B. The method is based on the fact that evaluating y = Tf(x)(x) is in general much

simpler (and faster) than evaluating y = f(x). On the other hand, if the expansion order is

properly selected, the function f is sufficiently regular, and the uncertainty set of x is not too

large, the error introduced by substituting f with its Taylor approximation is limited. Based on

these considerations, the DA-based MC method is a classical MC method in which the repetitive

evaluation of f is replaced by the evaluation of Tf : given a set of samples of x, the polynomial

Tf is evaluated repetitively and a statistical analysis is performed on the resulting y. It is worth

noting that, in the DA-based approach, the Taylor expansion Tf shall be computed only once, since

the same polynomial can be used in case new statistics need to be mapped. This is an additional

advantage with respect to classical MC, as already pointed out in [42].

D. Example: Cartesian to polar transformation

The problem of mapping a Gaussian pdf from Cartesian to polar coordinates is investigated

in this section for illustrative purposes. The section focuses on the full and partial DA-based pdf

mapping methods, since the application of the MC method is deemed to be straightforward. The

Cartesian to polar coordinate transformation is given by

y = f(x) =

(√x2

1 + x22

atanx2

x1

)(26)

and x is assumed to be a Gaussian random vector (GRV), with p(x) = N (µ,P ), in which

µ = (µ1, µ2) is the mean vector and P is a diagonal covariance matrix defined by the standard

deviations σ1 and σ2. The goal is to compute the mapped pdf p(y).

15

The analytical solution is available in this simple case. To write Eq. (14), the inverse transfor-

mation is

x = g(y) =

(y1 cos y2

y1 sin y2

)(27)

and the determinant of its Jacobian ∣∣∣∣det∂g∂y∣∣∣∣ = y1. (28)

Thus, based on Eq. (14), the mapped pdf can be written as

p(y) = N (µ,P )y1, (29)

which yields

p(y) =exp

(− (y1 cos(y2)−µ1)2

2σ21

− (y1 sin(y2)−µ2)2

2σ22

)y1

2πσ1σ2. (30)

Thus, p(y) is obtained explicitly in terms y: for any y, the evaluation of Eq. (30) supplies p(y) at

y.

The derivation of the inverse transformation of Eq. (27) can be avoided using Eq. (15), which

in this case reads

p(y) = p(f(x)) = N (µ,P )√x2

1 + x22, (31)

which makes use of the relation

1

/∣∣∣∣det∂f∂x∣∣∣∣ =

√x2

1 + x22. (32)

In its explicitly form, Eq. (31) reads

p(y) = p(f(x)) =exp

(− (x1−µ1)2

2σ21− (x2−µ2)2

2σ22

)√x2

1 + x22

2πσ1σ2. (33)

In this case, p(y) is obtained in terms of x: for any x, the evaluation of Eq. (33) supplies p(y) at

y = f(x).

Figure 2(a) shows the contour levels of p(x) for µ = (5, 4) and σ = (1, 1), in the range x ∈

µ± 32σ. In Figure 2(b), the contour levels of p(y) are plotted using the analytical expression (30)

or (33) equally.

16

(a) Initial pdf in cartesian variables (b) Mapped pdf in polar variables

(c) Contour levels of 1

/T∣∣∣det ∂f

∂x

∣∣∣(x) (d) Absolute error of 1

/T∣∣∣det ∂f

∂x

∣∣∣(x)

(e) Mapped pdf with the partial DA method (f) Absolute error of the partial DA method

(g) Mapped pdf with the full DA method (h) Absolute error of the full DA method

Fig. 2: Pdf mapping for the Cartesian to polar transformation: analytic solutions vs. Taylor ap-

proximation.

17

Figures 2(c) to 2(h) assess the accuracy of both the partial and full DA-based pdf mapping

methods using the particle-based method introduced in Sec. II C on a very fine grid for visualisation

purposes. The approach of Fig. 1(b) is adopted for the partial method. The vector x is initialized

as DA variable to compute the Taylor approximation of Eqs. (26) and (32) in the DA framework.

Then, for each x ∈ µ± 32σ, the resulting Taylor polynomials are evaluated to compute the approx-

imated values of both y and the inverse of the determinant of the Jacobian. The contour levels of

1

/T|det ∂f

∂x |(x) are plotted in Figure 2(c), whereas Figure 2(d) shows the absolute error of its 9-th

order Taylor approximation with respect to the analytical solution. The maximum absolute error

of the Taylor representation in the considered region is 3.84× 10−4. Once the Taylor expansion of

the inverse of the determinant of the Jacobian is available, its multiplication by p(x) supplies the

mapped pdf p(y). The result is shown in Figure 2(e). Clearly, the absolute error of the Taylor

approximation of p(y) with respect to the analytical solution, reported in Figure 2(f), has the same

behaviour of that of the Taylor representation of the inverse of the determinant of the Jacobian.

This is due to the fact that p(y) is obtained by multiplication with p(x), which is imposed and,

then, known analytically. The maximum absolute error of the partial DA method in the considered

region is 6.64× 10−6.

Figure 2(g) shows the contour levels of the mapped pdf obtained with the full DA method,

i.e. when also p(x) is approximated as a Taylor polynomial. The error of the mapped pdf in

this case is a combination of the error of the Taylor approximations of both p(x) and the inverse

of the determinant of the Jacobian. However, Figure 2(h) clearly shows that the error on the

polynomial representation of p(x) prevails: the absolute error is 4.67×10−1, resulting in a maximum

percentage error of 327.62%. The relative error becomes large when the pdf gets small because

the Taylor expansion fails to represent the asymptotic behaviour of p(x) independently of the

order used. Of course, the error would increase if a larger initial domain is considered, e.g. for

x ∈ µ ± 3σ. In this case, considering that the error is mainly due to the approximation of p(x)

which is axisymmetric, the truncation error of the full DA approach can be estimated by comparing

the value of p(x) and its DA approximation at any point at which the Mahalanobis distance is 3. In

this example, p(µ+ 3σ) = 1.7681× 10−3, whereas Tp(µ+ 3σ) = 1.3565. This result clearly shows

18

the intrinsic unsuitability of the DA approach when approximating an asymptotic function with a

single expansion. A single polynomial expansion is not sufficient to represent accurately p(x) and,

consequently, p(y) over the entire domain.

IV. Probabilistic IOD

A. Angles-only IOD

In the classical angles-only IOD problem, three optical observations at epoch ti, with i = 1, . . . , 3

are available. The observations consist of three couples of right ascension and declination angles

(αi, δi), or in a more compact notation (α, δ). These observations provide three inertial lines of

sight ρi, i.e. the unit vectors pointing from the observer (on the Earth’s surface) to the observed

object.

A method to use Taylor DA to solve and expand the solution of the corresponding IOD problem

has already been introduced by Armellin et al. [43]. This considers all the slant ranges ρi with

i = 1, . . . , 3 (in a more compact notation ρ) as the unknowns of the problem, for which initial

guesses are provided by a classical Gauss’ method. Starting from these initial guesses, two Lambert’s

problem, a first one between t1 and t2 and a second one between t2 and t3, can be solved to obtain

the two velocity vectors v−2 and v+2 . These two velocity vectors will be initially different as only

first guesses of the slant ranges are available. Thus, the IOD problem consists in finding the ρ

such that the residual function h(ρ) = v−2 − v+2 = 0. Typically just few Newton iterations are

sufficient to find the ρ that solves this set of three nonlinear equations [44]. Once ρ is known, the

position vectors ri can be readily computed; the associated velocity vectors are delivered by solving

the two Lambert problems. If the measurement angles are defined as DA numbers, the residuals

become functions of both the slant ranges and measurements angles, i.e. v−2 - v+2 = h(ρ,α, δ).

The goal is to find ρ = Tρ(α, δ), i.e. a Taylor approximation of the function that tells how the

slant ranges need to change to solve the IOD problem for any variation of the measurements within

their uncertainty domain. This is exactly the problem described in Sec. II B. When Tρ(α, δ) is

computed, the Tri(α, δ) are readily available. Finally, the Tvi(α, δ) are obtained by solving the

Lambert problems in the DA framework. The polynomials Tri and Tvican then be concatenated to

19

write the full state of the spacecraft at the epoch of the second observation as a Taylor polynomial

y2 = Ty2(α, δ).

For any value of α and δ belonging to their uncertainty domain, the evaluation of the polynomial

Ty2supplies a Taylor approximation of the state that solves the IOD problem. Evidently, this

Taylor polynomial is the approximation of the O2S map for the angles-only IOD, which supplies the

estimated spacecraft state for different values of the measurements. If the pdf of the observations,

i.e. p(α, δ), is known, the DA-based pdf mapping methods can be used to estimate the spacecraft

state pdf at t2. If the initial pdf is assumed to be Gaussian, the method described in Sec. III B can

be used to compute the statistical moments of the mapped pdf.

B. Two position vectors IOD

In the case of radar observations, two measurements at epoch ti, with i = 1, 2, are available.

Each observation consists of the pointing angles (αi, δi) and the range ρi, or in a more compact

form (α, δ,ρ). By assuming the observer’s location as known, each radar observation is equivalent

to measuring the spacecraft position vector. This greatly simplifies the IOD problem. If all αi, δi,

and ρi are initialized as DA numbers, simple algebraic operations allows the two position vectors

to be approximated in terms of their Taylor expansion with respect to the measurements, i.e. as

ri = Tri(ρi, αi, δi).

A Lambert problem is formulated with the resulting two position vectors and the elapsed time

between the observations. The Lambert problem can be solved in the DA framework as summarised

in Sec. II B 2 [45]. The result is the Taylor approximation of the velocity vectors at the two ob-

servation epochs, i.e. vi = Tvi(α, δ,ρ), for i = 1, 2. The polynomials Tri and Tvi

can then be

concatenated to write the full state of the spacecraft at the epoch of the first observation as a

Taylor polynomial y1 = Ty1(α, δ,ρ).

The polynomial Ty1is the Taylor expansion of the solution of the IOD problem with respect

to the measurements: the evaluation of the polynomial Ty1for any α, δ, and ρ delivers the state

y1 that solves the problem. Clearly, Ty1is the approximation of the O2S map for the two position

vectors IOD. Similarly to Sec. IVA, if the pdf of the observations, p(α, δ,ρ), is known, the DA-

20

based pdf mapping methods can be used to approximate the pdf of the estimated state at t1. If the

initial pdf is assumed to be Gaussian, the method described in Sec. III B can be used to compute

the statistical moments of the mapped pdf.

C. IOD in high-fidelity dynamics

The procedures for the angles-only IOD and the two position vectors IOD presented in the

previous sections rely on the assumption that the object moves in Keplerian dynamics. However,

both procedures can be extended to work for arbitrary dynamics. This section addresses such a

generalization, starting from the procedure for the angles-only IOD and then moving to the case of

the two position vectors.

Let us assume that a reference state y2 has been estimated in Keplerian dynamics from the

reference optical observations (αi, δi) at epoch ti, with i = 1, . . . , 3, or, with the compact notation,

from (α, δ). The generalization of the angles-only procedure goes through the following steps:

1. The state at epoch t2 and all the measurements are initialized as DA numbers, i.e. y2, α, and

δ, around their reference values.

2. Using simple algebraic operations, the measurements at t2 are computed from the state y2 to

obtain the polynomials αc2 = Tαc2(y2) and δc2 = Tδc2(y2).

3. The state vector y2 is propagated forward to t3 and backward to t1 in the dynamics of

interest using the DA version of the 7/8 Dormand-Prince integrator described in Sec. II A.

The propagations yield the Taylor expansions of the state vector at t1 and t3 with respect to

y2, i.e. y1 = Ty1(y2) and y3 = Ty3

(y2).

4. Similarly to step 2, the measurements at t1 and t3 are computed from y1 and y3, respectively,

to obtain the polynomials αci = Tαci(y2) and δci = Tδci (y2), with i = 1 and 3. Consequently,

the measurements at all epochs have been computed in terms of their Taylor expansion in y2.

With a compact notation, these polynomials are denoted as (αc, δc) = Tαc,δc(y2).

5. Clearly, the computed αc and δc must match α and δ. To impose this condition, the errors

∆α and ∆δ between (αc, δc) and (α, δ) are computed. By definition, these errors depend on

21

all y2, α, and δ:

(∆α,∆δ) = (αc −α, δc − δ) = (Tαc(y2)−α, Tδc(y2)− δ) = T∆α,∆δ(y2,α, δ). (34)

6. The polynomial in Eq. (34) can be inverted using the dedicated algorithm available in the DA

framework [25] to obtain:

y2 = Ty2(∆α,∆δ,α, δ). (35)

7. The condition (∆α,∆δ) = (0, 0) can now be imposed to obtain

y2 = Ty2|∆α,∆δ=0(α, δ), (36)

which is the Taylor expansion of the estimated state at t2 with respect to the measurements

(α, δ) at ti, with i = 1, . . . , 3.

The evaluation of the polynomial map in Eq. (36) at any (α, δ) delivers the corresponding

estimated state y2 in the dynamics used in step 3. Consequently, it is the Taylor expansion of the

O2S map in the dynamics of interest. Similarly to the previous sections, the pdf of the observations,

p(α, δ), can be mapped into y2 by applying the DA-based pdf mapping methods on Ty2|∆α,∆δ=0.

The above algorithm can be adapted to the two position vectors IOD. In this case, the reference

state y1 is obtained by solving the Lambert problem in the Keplerian dynamics between the position

vectors associated to the reference measurements (αi, δi, ρi) at epoch ti, with i = 1, 2, or, with the

compact notation, (α, δ, ρ). The generalization of the procedure of Sec. IVB for arbitrary dynamics

goes through the following steps:

1. The state at epoch t1 and all the measurements are initialized as DA numbers, i.e. y1, α, δ,

and ρ, around their reference values.

2. Using simple algebraic operations, the measurements at t1 are computed from the state y1 in

the DA framework: (αc1, δc1, ρ

c1) = Tαc

1,δc1,ρ

c1(y1).

3. The state vector y1 is propagated forward to the epoch t2 of the second observation in the

dynamics of interest. Once again, the propagation is carried out with the DA version of the

22

7/8 Dormand-Prince integrator. The propagation yields the Taylor expansion of the state

vector at t2 with respect to y1, i.e. y2 = Ty2(y1).

4. The measurements at t2 are computed from y2 to obtain the polynomials (αc2, δc2, ρ

c2) =

Tαc2,δ

c2,ρ

c2(y1). All the measurements have now been computed in terms of their Taylor ex-

pansions in y1, i.e. (αc, δc,ρc) = Tαc,δc,ρc(y1).

5. The computed (αc, δc,ρc) must match (α, δ,ρ). To impose this condition, the error between

the observables is obtained as

(∆α,∆δ,∆ρ) = (αc −α, δc − δ,ρc − ρ) = T∆α,∆β,∆ρ(y1,α, δ,ρ). (37)

6. The polynomial in Eq. (37) is inverted to obtain:

y1 = Ty1(∆α,∆δ,∆ρ,α, δ,ρ). (38)

7. The condition (∆α,∆δ,∆ρ) = (0, 0, 0) can now be imposed to obtain

y1 = Ty1|∆α,∆δ,∆ρ=0(α, δ,ρ). (39)

Similarly to Eq. (36), the polynomial map in Eq. (39) is the Taylor expansion of the estimated

state at t1 with respect to the measurements (α, δ,ρ) at t1 and t2, i.e. the O2S map of the two

position vector IOD problem. The map has been obtained in the general dynamics adopted in Step

3. Once the pdf of the measurements is available, this polynomial can be used to map it into y1

using the DA-based pdf mapping methods.

D. Arbitrary state representations

As detailed in the previous section, once the polynomial map of Eq. (36) or the one of Eq.

(39) is available, the DA-based pdf mapping methods described in Sec. IIIA can be used to map

the pdf of the measurements into the pdf of the estimated states. This is achieved by setting the

function f of Eq. (15) to Ty2|∆α,∆δ=0(α, δ) or Ty1|∆α,∆δ,∆ρ=0

(α, δ,ρ), respectively. Indeed, they

provide the transformation y = f(x) from the observations x to the state vector y in terms of its

Taylor expansion y = Tf (x).

23

Without loss of generality, let us assume that the above transformation has been derived by

representing the state vector y in a set of orbital parameters OP1. Thus, yOP1 = fOP1(x) and the

DA-based solution of the IOD problem has delivered its Taylor expansion yOP1 = TfOP1(x), which

is valid from the measurements x to the OP1 representation of y. It is worth observing that the

DA-based pdf mapping method works for y expressed in any set of orbital parameters, provided

that the necessary coordinate transformations are included in the function f . The coordinate

transformation from OP1 to any new set of orbital parameters OP2 is usually given by a nonlinear

function yOP2 = fOP1→OP2(yOP1), which consists of algebraic operations. The transformation

fOP1 can be composed with fOP1→OP2 to obtain the overall transformation

yOP2 = fOP2(x) = fOP1→OP2 fOP1(x) (40)

from the observations x to the OP2 representation of y. The evaluation of Eq. (40) in the DA

framework provides its Taylor expansion

yOP2 = TfOP2(x) = TfOP1→OP2

TfOP1(x), (41)

where TfOP1→OP2is the Taylor expansion of the coordinate transformation from OP1 to OP2.

Consequently, the DA-based pdf mapping methods can be applied on TfOP2to map the pdf of the

observations, p(x), into the pdf of the estimated state y represented in the arbitrary set of orbital

parameters OP2.

V. Test Cases

Single-pass optical and radar observations of the two objects listed in Table 1 are considered

as test cases for the DA-based pdf mapping methods. The first one is INTELSAT 901, a GEO

communications spacecraft. The second one is an SL-8 rocket body in LEO.

24

Table 1: Test cases: orbital parameters

Test Case A B

Orbit type GEO LEO

SSC 26824 04784

Epoch JED 2457163.2824 2457155.9737

a km 42143.781 7353.500

e – 0.000226 0.699849

i deg 0.0356 74.0295

Ω deg 26.278 179.640

ω deg 42.052 359.079

M deg 72.455 13.735

For the object in GEO, an optical campaign is simulated with three observations taken with the

European Space Agency Optical Ground Station (OGS, Teide Observatory, Canary Islands). The

observations are separated by 12 min (approximately by an arc length of 3 deg). A resolution of

σα,δ = 0.5 arcsec is considered for both the right ascension and declination.

For the LEO case, two radar observations taken with the Chilbolton Advanced Meteorological

Radar (CAMRa, Chilbolton Observatory, UK) are simulated. The observations are separated by 2

minutes, approximately an arc length of 7 deg, such that the range from the station remains of the

order of 1000 km. The range has a resolution of σρ = 25 m, and the pointing angles have standard

deviations σα,δ = 0.25 deg.

In both campaigns, the observations are simulated using the following procedure. First of all,

a set of measurements is generated using the orbital parameters of Table 1, without adding any

noise. As solving the IOD problem with these measurements would deliver the true solutions (i.e.

the orbital parameters of Table 1), these measurements are referred to as true measurements. Then,

the simulated optical and radar observations are generated by adding a random error to the true

measurements, using the standard deviations introduced above. The resulting observations are

labelled as nominal as they are the reference observations in the IOD solution processes described

25

in Sec. IV; i.e. all the Taylor expansions are computed around them. The nominal observations

are a particular realization of the noisy measurements, and thus differ from the true observations.

Accordingly, the nominal solution of the IOD process will differ from its true solution. To highlight

this difference the nominal and the true solutions will be indicated in the analysis of the results with

a diamond and a square marker, respectively. Unless differently specified, the results were obtained

using Taylor expansions at 6-th order. This value was selected based on the accuracy analyses

reported in the discussion of the results.

A. Test Case A

Figure 3 shows the results of the 6-th dimensional pdf mapped from the observation space

into the modified equinoctial elements space yMEE = (p, f, g, h, k, L), in which p = a(1− e2) is the

semilatus rectum, f = e cos(ω+Ω) and g = e sin(ω+Ω), h = tan(i/2) cos(Ω) and k = tan(i/2) sin(Ω),

and the true longitude L = Ω+ω+ν. The set yCOE = (a, e, i,Ω, ω, ν) represents instead the classical

orbital elements. The two-body dynamics is used in this case. The plots are obtained by processing

100,000 normally distributed vectors generated in the observation space using the MATLAB function

mvnrnd. Each sample is mapped from the observation space (α, δ) into the state space yMEE at

the epoch of the second observation by evaluating the Taylor approximation yMEE = TyMEE(α, δ),

i.e. the straightforward application of the DA-based MC method described in Sec. III C.

As explained in Sec. III A, the pdf of each observation sample is obtained using the partial DA-

based pdf mapping method, i.e. by multiplying the pdf p(α, δ) of the measurements by the Taylor

representation of the inverse of the determinant of the Jacobian of the O2S map, i.e. p(yMEE) =

p(TyMEE(α, δ)) = p(α, δ) 1/

∣∣∣∣T ∂yMEE∂(α,δ)

∣∣∣∣. The colour of each sample is based on the value assumed

by the pdf, which has been scaled in the range [0, 1] for a better readability of the plots.

The figures on the left column of Fig. 3 show the projection of the pdf on the (p, L), (f, g),

and (h, k) planes. In these plots, the true (square marker) and the nominal (diamond marker)

IOD solutions are reported. As anticipated at the beginning of Sec. V, the true solution differs

from the nominal solution of the probabilistic IOD, since the nominal observations are a particular

realization of the noisy measurements. On the other hand, as expected, the true solution belongs

26

to the uncertainty set resulting from the probabilistic IOD process.

It is worth mentioning that, unlike an MC simulation, in which one needs to evaluate the

O2S map in many samples to be able to perform a meaningful statistical analysis of the results,

the samples are here used only for visualisation purposes. In fact, by using the partial DA-based

approach, the pdf of each sample can be directly computed by evaluating the associated analytical

functions, without resorting to a statistical analysis performed on a set of samples.

27

(a) (b)

(c) (d)

(e) (f)

Fig. 3: Mapped pdf in the MEE space for optical observations

Figure 3 shows that, although the size of the uncertainty map is large, the mapped pdf still

(visually) resembles a Gaussian distribution. The uncertainty set is large as the optical observations

are taken on a short arc, and, thus, the sensitivity of the solution to the measurements is high.

In addition, in a short arc, the nonlinearities of the dynamics do not play a significant role and,

28

thus, the pdf remains nearly Gaussian. This qualitative argument is confirmed by comparing the

covariance-correlation matrices computed at 6-th and 1-st order, which are reported in Table 2 and

3 respectively. The difference between the two matrices in all the components is less than 1%. These

matrices were computed with the approach described in Sec. III B, i.e. without using samples.

p [km] f g h k L [rad]

7.9707e+05 -8.4471e+00 -1.1551e+01 1.6564e-01 -1.1906e-01 -2.3042e-02

-9.9701e-01 9.0058e-05 1.2202e-04 -1.7723e-06 1.2389e-06 2.4297e-07

-9.9913e-01 9.9297e-01 1.6768e-04 -2.3883e-06 1.7422e-06 3.3485e-07

9.9128e-01 -9.9783e-01 -9.8543e-01 3.5029e-08 -2.3938e-08 -4.7516e-09

-9.6999e-01 9.4962e-01 9.7864e-01 -9.3033e-01 1.8901e-08 3.4957e-09

-9.9659e-01 9.8867e-01 9.9850e-01 -9.8033e-01 9.8184e-01 6.7066e-10

Table 2: Covariance values between states (upper triangle), correlation values (lower triangle), and

variances (diagonal terms) computed at 6-th order for optical observations.

p [km] f g h k L [rad]

7.9373e+05 -8.4153e+00 -1.1508e+01 1.6525e-01 -1.1877e-01 -2.2987e-02

-9.9701e-01 8.9756e-05 1.2161e-04 -1.7687e-06 1.2364e-06 2.4248e-07

-9.9914e-01 9.9295e-01 1.6712e-04 -2.3835e-06 1.7387e-06 3.3417e-07

9.9146e-01 -9.9795e-01 -9.8553e-01 3.4998e-08 -2.3915e-08 -4.7472e-09

-9.7014e-01 9.4966e-01 9.7874e-01 -9.3026e-01 1.8884e-08 3.4925e-09

-9.9678e-01 9.8877e-01 9.9862e-01 -9.8032e-01 9.8182e-01 6.7005e-10

Table 3: Covariance values between states (upper triangle), correlation values (lower triangle), and

variances (diagonal terms) computed at 1-st order for optical observations.

It is important to notice that the selection of MEE is particularly suited to maintain the normal-

ity of the statistics. The effect of selecting different orbital elements representations on the mapped

pdf is shown in Fig. 4(a), in which the dimensionality of the pdf is reduced to 2 by binning in the

semi-major axis – eccentricity space. Since the orbital eccentricity cannot take negative values, the

mapped pdf is clearly visually non Gaussian when classical orbital elements are used. This is fur-

29

ther highlighted by the normality plots in Fig. 5. In particular, Figure 5(f) shows that also the true

anomaly is not normally distributed. This is due to both the low eccentricity and low inclination

of the test orbit: a small variation in the observations can produce a large variation in all the true

anomaly, the pericenter anomaly and the right ascension of the node. The true longitude (the sum

of these three angles) is significantly less affected.

(a) Samples binned in the semi-major axis – eccen-

tricity space

(b) Mapped pdf in the semilatus rectum – true longi-

tude space for uniform distribution

Fig. 4: Analysis of the impact of state representation and initial pdf on the mapped pdf for optical

observations.

30

(a) (b)

(c) (d)

(e) (f)

Fig. 5: Normality plots for optical observations: comparison between MEE (left) and COE (right).

The dashed lines are extensions of the line connecting the 25th and 75th percentiles of the dis-

tribution: proximity of the distributions to the dashed lines indicates validity of the normality

assumption.

The proposed DA-based pdf mapping method is independent of the particular pdf adopted for

31

the observations. Figure 4(b) illustrates the result of mapping a pdf that is uniformly distributed

in the 6-th dimensional cube of side [−3σα,δ, 3σα,δ] in the observation space. In this particular

case, the mapped pdf depends only on the determinant of the Jacobian of the transformation, since

p(α, δ) is constant over the entire cube.

The proposed mapping method can be used to map a pdf to any arbitrary time. To this

aim, suppose the Taylor expansion of the O2S map has been obtained at the epoch of the second

observation t2 using the techniques introduced in Sec. IV:

y2 = Ty2(α, δ). (42)

The initial condition y2 can be propagated forward to any future epoch tf using the DA solution of

the Kepler problem when two-body dynamics is considerd or the DA version of the 7/8 Dormand-

Prince integrator for an arbitrary dynamical model. In either case the result is the Taylor expansion

of the final state yf with respect to the measurements α and δ taken at t2, i.e.

yf = Tyf(α, δ). (43)

It is worth observing that the polynomial map Tyfincludes the dependencies on α and δ deriving

from the solution of the IOD problem. Thus, given any specific set of observations α and δ, the

evaluation of Eq. (43) is an approximation of the result one would obtain by solving the IOD

problem at t0 and propagating the estimated state to tf in cascade. Similarly to Eq. (36) and Eq.

(39), the polynomial Tyfcan be used to map p(α, δ) into the pdf of yf using the DA-based pdf

mapping methods. To show the results of this procedure, Fig. 10 reports the 100,000 normally

distributed samples introduced above after binning them in the space of the semilatus rectum and

true longitude. On the left side the bins are plotted at the epoch of the second observation t2,

whereas on the right side a 6-th order expansion of the solution of Kepler’s problem is used to map

them ahead for 1 day. It is apparent how the dynamics stretches the set along the orbit. After

approximately one revolution the set is spread on an arc length of more than 50 deg. This result

is supported also by the covariance-correlation matrix reported in Table 4, which shows an increase

of four orders of magnitude in the σL with respect to Table 2. Note that, as two body dynamics is

assumed, this matrix correctly differs from the one at t2 in the last column and row only.

32

(a) Epoch t2 (b) Epoch t2+ 1 day

Fig. 6: Samples binned in the semilatus rectum – true longitude space for optical observations at

different epochs.

p [km] f g h k L [rad]

7.9707e+05 -8.4471e+00 -1.1551e+01 1.6564e-01 -1.1906e-01 -1.7035e+02

-9.9701e-01 9.0058e-05 1.2202e-04 -1.7723e-06 1.2389e-06 1.8053e-03

-9.9913e-01 9.9297e-01 1.6768e-04 -2.3883e-06 1.7422e-06 2.4685e-03

9.9128e-01 -9.9783e-01 -9.8543e-01 3.5029e-08 -2.3938e-08 -3.5391e-05

-9.6999e-01 9.4962e-01 9.7864e-01 -9.3033e-01 1.8901e-08 2.5433e-05

-9.9983e-01 9.9683e-01 9.9887e-01 -9.9085e-01 9.6935e-01 3.6420e-02

Table 4: Covariance values between states (upper triangle), correlation values (lower triangle), and

variances (diagonal terms) computed at 6-th order after one revolution for optical observations.

The proposed methods are valid as long as the Taylor approximation of the transformations is

accurate enough. The coefficient-based method presented in Sec. II C was first used to verify that

the truncation error of the 6-th order approximation of the O2S function was sufficiently small in the

domain of interest. The results of this analysis are reported in Fig. 7(a), showing that the absolute

errors are sufficiently small in the 6-th dimensional cube of side [−3σα,δ, 3σα,δ]. However, in an IOD

problem, it is more important to verify whether the predicted observations computed from the Taylor

approximation of the state Ty2(α, δ) are close enough to the real observations, for any admissible

33

value of the measurements noise. In other words, we need to verify that the observational residuals

derived from the evaluation of the Taylor approximation of the O2S function are sufficiently small.

To verify this, samples are generated in the observation space (referred to as sampled observations)

and the associated IOD solution is computed using the polynomial map. Then, each IOD solution is

used to generate the predicted observations by evaluating the measurement function. The accuracy

of the expansion is assessed by a statistical analysis of the difference between the sampled and

the predicted observations, relative to the given perturbation; i.e. the relative residuals. This is a

sample-based approach for checking the accuracy of the expansion (according to the classification

proposed in Sec. II C), but it has the advantage of not requiring multiple pointwise solutions of the

IOD problem, which would be time consuming. In this approach the accuracy analysis is focused on

the observational residuals, which is what matters in the IOD problem: any state with sufficiently

small residuals shall be regarded as a solution of the IOD problem. Figure 7(b) shows the empirical

cumulative distribution function of the relative residuals for different expansion orders. The relative

error decreases with the expansion order and a 6-th order computation guarantees a maximum

relative error of the order of 1× 10−3, corresponding to maximum residual of 2.0771× 10−3 arcsec

(1.0070 × 10−8 rad). This analysis clearly shows that the 6-th order expansion Ty2(α, δ) can be

regarded as an accurate approximation of the O2S function in the domain of interest.

(a) Estimation of the truncation error of Ty2(α, δ) (b) Empirical cumulative distribution function of the

relative residuals

Fig. 7: Analysis of the accuracy of the DA method at different expansion orders for optical obser-

vations.

34

B. Test case B

The results obtained for the radar observations are analysed with the same approach as for test

case A, using again 100,000 normally distributed samples in the observations space. Figure 8 shows

the samples as well as their probability, scaled in the range [0, 1]. In this case the large uncertainties

of the instrument pointing angles yield a large uncertainty set in the MEE space. Nevertheless, the

normality distribution of the samples is preserved due to the short temporal separation between the

observations.

35

(a) (b)

(c) (d)

(e) (f)

Fig. 8: Mapped pdf in the MEE space for radar observations

This is clearly confirmed by the analysis of the covariance matrices obtained with 6-th and 1-st

order computations, as reported in Table 5 and 6. However, similarly to the angles-only IOD, the 6-

th order expansion is significantly more accurate in representing the O2S map. This is shown in Fig.

9, in which the same analyses of the angles-only case are reported. The estimated truncation errors

36

of the state is much smaller for radar observations, as the associated IOD problem is simpler due to

the full observation of two position vectors. This is confirmed by Fig. 9(b), in which the empirical

cumulative distribution function of the relative residuals is plotted. For the 6-th order expansion,

the relative residual is of the order of 1×10−7, corresponding to a maximum residual of 2.7441×10−3

arcsec on the angles and 2.1326× 10−6 km on the range. As these values are much smaller than the

instrument resolution we can conclude that Ty1(α, δ, ρ) is an accurate approximation of the O2S

function in the domain of interest.

p [km] f g h k L [rad]

7.7266e+03 -6.7672e-01 -8.2801e-01 -8.4973e-02 -1.0769e-01 6.4272e-02

-9.6537e-01 6.3598e-05 6.9396e-05 4.8983e-06 6.2119e-06 -1.7431e-06

-9.8752e-01 9.1225e-01 9.0989e-05 1.0903e-05 1.3797e-05 -9.6279e-06

-3.7202e-01 2.3637e-01 4.3988e-01 6.7522e-06 9.6472e-06 -9.2053e-06

-3.2775e-01 2.0838e-01 3.8694e-01 9.9317e-01 1.3973e-05 -1.3320e-05

2.0246e-01 -6.0520e-02 -2.7947e-01 -9.8088e-01 -9.8667e-01 1.3044e-05

Table 5: Covariance values between states (upper triangle), correlation values (lower triangle), and

variances (diagonal terms) computed at 6-th order for radar observations.

p [km] f g h k L [rad]

7.7271e+03 -6.7678e-01 -8.2807e-01 -8.4976e-02 -1.0770e-01 6.4277e-02

-9.6540e-01 6.3600e-05 6.9404e-05 4.8979e-06 6.2123e-06 -1.7430e-06

-9.8753e-01 9.1233e-01 9.0994e-05 1.0904e-05 1.3799e-05 -9.6289e-06

-3.7203e-01 2.3636e-01 4.3992e-01 6.7516e-06 9.6467e-06 -9.2047e-06

-3.2778e-01 2.0839e-01 3.8698e-01 9.9319e-01 1.3973e-05 -1.3320e-05

2.0247e-01 -6.0516e-02 -2.7950e-01 -9.8089e-01 -9.8667e-01 1.3043e-05

Table 6: Covariance values between states (upper triangle), correlation values (lower triangle), and

variances (diagonal terms) computed at 1-st order for radar observations.

37

(a) Estimation of the truncation error of Ty2(α, δ,ρ) (b) Empirical cumulative distribution function of the

relative residuals

Fig. 9: Analysis of the accuracy of the DA method at different expansion orders for radar observa-

tions.

As for the optical case, the selection of the set of orbital parameters used for the representation

of the spacecraft state plays a key role on the characteristics of the mapped pdf. By comparing Fig.

10(a) with Fig. 10(b), it is clear that, when classical orbital elements are used, the samples are no

longer normally distributed in eccentricity, since this orbital parameter cannot take negative values.

(a) Samples binned in the semi-major axis and eccen-

tricity space at t1

(b) Samples binned in the true longitude and semilatus

rectum space at t1

Fig. 10: Effect of state representation on the mapped pdf for radar observations.

As already mentioned in Sec. IVB, given the two position vectors and the elapsed time between

the observations in the radar observations scenario, a Lambert problem can be solved to compute

38

the Taylor approximation of the velocity vectors at the two observation epochs using DA. The

solution of the Lambert problem relies on the assumption of a Keplerian motion. Nevertheless, the

proposed method can be easily extended to apply to more complex dynamics as described in Sec

IVC. This approach is used here to solve the IOD problem by predicting the motion of the object

using the numerical propagator AIDA (Accurate Integrator for Debris Analysis [46]), which includes

geopotential acceleration, atmospheric drag, solar radiation pressure, and third body gravity as

perturbations to the Keplerian motion.

Table 7 compares the mean MEEs at the epoch of the first observation obtained using Kepler’s

dynamics with the MEEs at the same epoch computed with AIDA. Due to the short time between

the two observations, the effect of the perturbations is negligible. This is confirmed by comparing

the covariance-correlation matrices of the MEE samples obtained with 6-th order computations in

Keplerian dynamics (Table 5) with the one reported in Table 8 and obtained using AIDA.

p [km] f g h k L [rad]

Kepler 7.2699e+03 5.9892e-03 8.0798e-03 -7.5380e-01 4.7789e-03 1.0305e+01

AIDA 7.2708e+03 5.9534e-03 7.9465e-03 -7.5381e-01 4.76018e-03 1.03049e+01

Table 7: Comparison between the mean MEE computed at 6-th order with Keplerian dynamics and

AIDA model.

39

p [km] f g h k L [rad]

7.7278e+03 -6.7678e-01 -8.2818e-01 -8.4972e-02 -1.0770e-01 6.4270e-02

-9.6536e-01 6.3600e-05 6.9404e-05 4.8965e-06 6.2096e-06 -1.7406e-06

-9.8752e-01 9.1223e-01 9.1011e-05 1.0904e-05 1.3798e-05 -9.6284e-06

-3.7202e-01 2.3630e-01 4.3988e-01 6.7510e-06 9.6459e-06 -9.2040e-06

-3.2775e-01 2.0830e-01 3.8694e-01 9.9317e-01 1.3972e-05 -1.3319e-05

2.0244e-01 -6.0437e-02 -2.7947e-01 -9.8088e-01 -9.8666e-01 1.3042e-05

Table 8: Covariance values between states (upper triangle), correlation values (lower triangle), and

variances (diagonal terms) computed at 6-th order when the radar IOD problem is solved with

AIDA dynamical model.

The same approach adopted to derive Eq. (43) in test case A can be used here to map the pdf of

the measurements at epoch t1 into the pdf of the state at epoch t1+T , in which T is the period of the

nominal solution. Unlike the previous case, to investigate the long-term effects of the perturbations

and show the versatility of the proposed method, the pdf is mapped by computing the 6-th order

expansion of the flow using AIDA. The result is shown in Figure 11(a) by binning the resulting

samples. Once again, the dynamics stretch the set along the orbit and, after one revolution, the

set is spread on an arc length of about 30 deg. As a clear effect of the increasing nonlinearities,

the set starts bending, so drifting apart from the normal distribution. This result is quantified by

the covariance of the propagated samples reported in Table 9, in which the variance of L grows

by three orders of magnitude. It is worth to observe that, in contrast with the two-body model,

all the elements of the covariance matrices are time varying due to the effect of the perturbations.

Lastly, Fig. 11(b) reports the predicted truncation error of Tyf(α, δ,ρ). Due to the large initial

domain the accuracy of the Taylor representation of the true longitude quickly degrades, and after

one revolution the truncation error at 6-th order increases by six orders of magnitude, passing from

1.52× 10−12 to 2.04× 10−6 rad. What is more important to note is that increasing the expansion

order has a small effect on the truncation error, a situation that can be alleviated only by suitably

splitting the initial domain [38].

40

(a) Samples binned in the true longitude and semila-

tus rectum space after one orbital revolution

(b) Estimation of the truncation error of Tyf(α, δ,ρ)

Fig. 11: Mapped pdf and accuracy analysis of the approximation of the O2S function for radar

observations when the solution is propagated forward for one orbit revolution using AIDA dynamical

model.

p [km] f g h k L [rad]

7.7289e+03 -6.7687e-01 -8.2827e-01 -8.4975e-02 -1.0769e-01 -9.5712e+00

-9.6516e-01 6.3636e-05 6.9392e-05 4.9001e-06 6.2141e-06 8.4213e-04

-9.8744e-01 9.1171e-01 9.1033e-05 1.0902e-05 1.3795e-05 1.0230e-03

-3.7197e-01 2.3639e-01 4.3973e-01 6.7523e-06 9.6473e-06 9.6687e-05

-3.2770e-01 2.0839e-01 3.8680e-01 9.9317e-01 1.3974e-05 1.2089e-04

-9.9900e-01 9.6869e-01 9.8382e-01 3.4143e-01 2.9676e-01 1.1876e-02

Table 9: Covariance values between states (upper triangle), correlation values (lower triangle), and

variances (diagonal terms) computed at 6-th order for the radar IOD problem when the solution is

propagated with the AIDA dynamical model for one revolution.

C. Computational Times

The simulations presented in this work were run on a MacBook Air with a 1.8 GHz Intel Core i5

processor and 4 GB 1600 MHz memory. In Table 10 a comparison between the CPU times required

for a DA-based MC and a pointwise MC is provided for both Keplerian and AIDA dynamical models.

41

As expected both the optical and radar 6-th order solutions require more CPU time with respect to

a classical pointwise solution. In addition, the optical IOD problem is more time consuming than

the radar one, as the latter requires only the solution of one Lambert problem. However, the time

required for the evaluation of a 6-th order map is always (independently from the dynamical system

used) lower than a pointwise solution. As a result, a DA-based MC analysis is in all cases more

efficient when the number of samples to be evaluated is greater than 312. The situation becomes

much more in favour of the DA approach when a more complex dynamical system is used as shown

by the AIDA column in the table. It is worth noting that the availability of the DA expansion of the

O2S function enables the computation of both the mapped pdf and its statistical moments without

samples.

Keplerian AIDA

Optical Radar Radar

DA-based IOD solution 1.186 6.295 × 10−2 31.994

Single DA map evaluation 1.943 × 10−4 9.046 × 10−5 3.225 × 10−4

Pointwise IOD solution 4.001 × 10−3 3.110 × 10−4 0.564

MC break even point 312 286 57

Table 10: Comparison between the CPU times (in seconds) between 6-th order DA computations

and pointwise computations.

VI. Conclusions

A general method for mapping a pdf from the observations space into the object orbital space

in the frame of the solution of an IOD problem has been presented. This method is based on the

high order Taylor expansion of both the O2S map and its Jacobian obtained using DA techniques.

A simple Cartesian to polar transformation has been used as illustrative example to show how the

method works in a problem in which the analytical solution is available. Then, the approach was

run on two IOD scenarios: an optical observation of a spacecraft in GEO and a radar observation

of a spent rocket in LEO. It was shown that DA can be effectively used to nonlinearly map pdfs

(and compute their statistical moments), thus enabling a probabilistic analysis of IOD problems

42

without the need of samples. The availability of the Taylor expansion of the O2S map enables

also the execution of DA-based MC simulations. These are in general more efficient than classical

MC, but one needs to accept a loss in accuracy. However, it was shown that observation residuals

associated with 6-th order computations were orders of magnitude smaller than the resolution of

both optical and radar instruments. Finally, the method is independent of the model used to describe

the spacecraft dynamics, as well as of the pdf adopted for the measurements. For these reasons the

proposed approach has the potential to be used in data association problems, where the capability

of attributing a statistical meaning to the IOD solutions is a key.

Acknowledgments

R. Armellin acknowledges the support received by the Marie Sklodowska-Curie grant 627111

(HOPT - Merging Lie perturbation theory and Taylor Differential algebra to address space debris

challenges). This work has been partially funded by the Spanish Finance and Competitiveness

Ministry under Project ESP2014-57071-R.

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